Estimation of Ideal Points Using Cast Vote Records

Quant III and Bayesian Measurement

Mason Reece

December 4, 2023

Introduction

How representative is a government of its people?

  • One approach (of many) compares the “ideological” positions of legislators and voters

Cast Vote Records

  • Anonymous records of what candidate each voter picked in each race on the ballot

  • Modelling challenges

    • Categorical outcome with different options for each race

    • People vote in different races

  • 1.3B “bubbles” and 50m voters in 2020 election

Distribution of Cast Vote Records

Methodology

Continuous, latent outcome with categorical indicators –> IRT model

\[\begin{align*} Y_{j, k(c)} & \sim Bernoulli(\pi_{j, k(c)}) \text{ or } Categorical(\pi_{j, k(c)}) \\ \pi_{j, k(c)} &= logit^{-1} ( \nu_{j, k(c)} ) \text{ or } softmax( \nu_{j, k(c)} ) \\ \nu_{j, k(c)} &= \gamma_{k(c)} \alpha_j - \beta_{k(c)} \end{align*}\]
Quantity Symbol
Individual \(j = 1, 2, …, J\)
Race \(k = 1, 2, …, K\)
Candidate \(c = 1, 2, …, C\)
Ideal point of voter \(j\) \(\alpha_j\)
Discrimination/Slope Parameter \(\gamma_{k(c)}\)
Difficulty/Location Parameter \(\beta_{k(c)}\)

Identification and Priors

  • The scale and location of \(\alpha\) are unidentified (can add or multiply a constant with no change in likelihood)

    • Set prior to \(\mathcal{N}(0, 1)\)

  • \(\gamma\)’s sign can also flip around with relation to the sign of \(\alpha\).

    • Constrain \(\gamma > 0\) for Bernoulli models and post-process for Categorical models

  • Priors

    • Uninformative, standard priors for other parameters

    • Future work: Informative priors using party or NOMINATE/DIME scores to help with identification and speed convergence

Convergence

Bernoulli Results - \(\alpha\)

Binary Results - \(\beta\) and \(\gamma\)

Categorical Results – \(\alpha\)

Categorical Results – \(\alpha\)

Categorical Results - \(\gamma\)

Categorical Results - \(\beta\)

Future Work

  • Informative priors from DIME or NOMINATE scores

  • Multiple dimensions

  • Connect this better to a question about representation

Reece – CVR Ideals